# Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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### How to define the sum of cardinals in the definition of an inaccessible cardinal?

I'm reading the Wikipedia definition of an inaccessible cardinal and I'm trying to understand it. On Wikipedia, a (strongly) inaccessible cardinal $\kappa$ is defined in the following way: $\kappa$ ...
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### An infinite linear system of equations with an uncountable number $A$ of equations

I will start with an example to make things clear and avoid confusion : Take all $x>0$ and $$\exp(x) = \sum_{-1<n} a_n x^n$$ Now finding $a_n$ can be described as an infinite linear system of ...
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### $H_\kappa = \{A: |TC(A)|<\kappa \} = \{A: (\forall x \in TC(\{A\})) (|x|<\kappa)\}$ for singular cardinal $\kappa$

$TC(A)$ is the transitive closure of $A$, $|x|$ is the cardinality of $x$, $\kappa$ is a cardinal number greater than $\aleph_0$ and $H_\kappa$ is the set of sets hereditarily of cardinality less than ...
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### Can this statement be proved without the Continuum hypothesis?

I was just learning about set theory and came across this exercise. It does not specify whether we work with CH as an axiom. The statement is easy to prove if CH is assumed, but I'm not sure where to ...
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### Cardinality of bijection of $\mathbb{N} \rightarrow \mathbb{N}$ using finite sets [duplicate]

I was solving the following exercise : Is $\mathfrak{S}(\mathbb{N})$ countable ? Here $\mathfrak{S}$ stands for the set of bijections from $\mathbb{N}$ to itself. A proof could be to build an ...
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### Dimension of function space $X \to \mathbb F$

Let $X$ be an arbitrary set, and let $\mathbb F$ be an arbitrary field. Functions $X \to \mathbb F$ form a vector space over $\mathbb F$, with pointwise addition and scalar multiplication. What is its ...
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### Infinite recursion of function defined with respect to "every $n$" [closed]

I have very little formal mathematical training, so I apologize in advance for what may seem like basic issues in this question's phrasing. I am trying to determine whether I can make a certain ...
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### Category Theory and Cardinality

This is a very basic question from Rotman's Homological Algebra text. I think the issue is that I don't know how to rigorously talk about cardinality. Problem If $\mathcal A$ is a small category (...
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### Why is the cardinality of irrationals greater than of rationals?

If the rationals are dense in the rationals, meaning there is a rational between every 2 irrationals, then there is essentially a rational number for every irrational. If there is aleph irrationals, ...
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### How to describe a function that depends on the cardinality of a vector with mathematical notation

I have to describe a code I have written using formal mathematical notation, but I don't know how to define an operation that depends on the number of elements. Here's an example pseudocode: ...
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### There is no surjective function from a set in the Hartogs number of its power set

I'm trying to prove an equivalent state of the Axiom of Choice : Given two non-empty sets, there is a surjective function from one of them into the other one. If we prove that for any non-empty set $X$...
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### Reference request for different definitions of finite

I understand that there are different definitions of finite. They are all equivalent over ZFC, and some of them are even equivalent over ZF, but not over weaker theories than ZF(C). I would like some ...
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### Cardinality of union of sets of a given cardinality

I know that: finite union of finite sets is a finite set countable union of countable sets is a countable set It is possible to characterize the sets $C$ of cardinals satisfying the following ...